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Equation of a Circle Calculator

Select the generic form and enter the required inputs. The calculator will convert it to the standard form of the circle equation, determine its key parameters, and display a graphical representation of the result.

Equation Form:

Standard Form: (x - A)² + (y - B)² = C

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Use our Equation of a Circle Calculator to quickly find the equation of any circle and its key parameters. This tool helps you calculate radius, center, circumference, area, and even graph the circle for better visualization. Perfect for students, teachers, and geometry enthusiasts!

What Is a Circle?

In geometry:

“A circle is a set of points in a plane that are all equidistant from a fixed point called the center.”

equation of circle

Equation of a Circle

The general equation describes all points on a circle:

$$ (x-h)^2 + (y-k)^2 = r^2 $$

Where:

(h, k) = coordinates of the circle’s center
r = radius of the circle

Standard Form

If the center is at the origin (0, 0), the equation simplifies to:

$$ x^2 + y^2 = r^2 $$

Our circle equation calculator can automatically convert to this standard form.

Parametric Form

The parametric form uses an angle parameter to represent points on the circle:

x = r * cos(θ), y = r * sin(θ)

Where:

x and y = coordinates of a point on the circle
θ = angle measured from the center
r = radius

If the center is (h, k), the parametric form becomes:

x = h + r * cos(θ), y = k + r * sin(θ)

Conversion Between Parametric and Standard Form

Using the identity sin²(θ) + cos²(θ) = 1, substituting cos(θ) = (x-h)/r and sin(θ) = (y-k)/r gives:

$$ \left(\frac{x-h}{r}\right)^2 + \left(\frac{y-k}{r}\right)^2 = 1 $$

Multiplying through by r²:

$$ (x-h)^2 + (y-k)^2 = r^2 $$

Alternative (General) Form

The expanded form of a circle equation is:

$$ x^2 + y^2 + Dx + Ey + F = 0 $$

This can be calculated directly with the circle equation calculator.

Circumference

The total distance around the circle is:

$$ C = 2\pi r $$

Use our circumference calculator for quick results.

Area

The area enclosed by the circle is:

$$ A = \pi r^2 $$

Find the area of sectors with our area of a sector calculator.

Eccentricity

For a circle, eccentricity is 0 because it is a special case of an ellipse with coinciding foci.

Domain and Range

Domain: $$ [h-r, h+r] $$

Range: $$ [k-r, k+r] $$

Example Calculations

Example 1: Find the equation of a circle with center (3, 7) and radius 3.

$$ (x-3)^2 + (y-7)^2 = 3^2 $$

$$ (x-3)^2 + (y-7)^2 = 9 $$

Example 2: Find the center of the circle given by:

$$ (x-4)^2 + (y-2)^2 = 25 $$

Comparing with standard form: center = (h, k) = (4, 2), radius = 5

Expanded form:

$$ x^2 + y^2 - 8x - 4y - 5 = 0 $$

How the Circle Equation Calculator Works

Input:

Select parameters from the drop-down menu
Enter values for center, radius, or general form coefficients
Click calculate

Output:

Standard and general forms
Radius and diameter
Area and circumference
Domain and range
Eccentricity
Center coordinates
Parametric form
Graph of the circle

FAQs

What is the circle theorem?

It states that if a triangle has one side as the diameter, then the angle opposite this side is a right angle.

What is a secant?

A secant is a line intersecting the circle at exactly two points.

Are all radii equal?

Yes, every radius has the same length from the center to the circumference.

Find the radius of $$ (x-5)^2 + (y-8)^2 = 36 $$

Radius = 6

Do all chords pass through the center?

No. Only diameters pass through the center; all diameters are chords, but not all chords are diameters.

Conclusion

The circle equation helps calculate area, circumference, and other key properties, and it is widely used in geometry, engineering, and astronomy. Using the circle equation calculator makes these computations fast and accurate.

References

From Wikipedia: Circle. From Khan Academy: Circle equation review. From Lumen Learning: The Circle and the Ellipse.

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